"Frontmatter". In: Analysis of Financial Time Series

DURATION MODELS 199actions might not be constant or monotone over time. To increase the flexibility of theassociated hazard function, Zhang, Russell, and Tsay (2001) employ a (standardized)generalized Gamma distribution for ɛ i . See Appendix A for some basic properties ofa generalized Gamma distribution. The resulting hazard function may assume variouspatterns, including U shape or inverted U shape. We refer to an ACD model withinnovations that follow a generalized Gamma distribution as a GACD(r, s) model.5.5.2 SimulationTo illustrate ACD processes, we generated 500 observations from the ACD(1, 1)modelx i = ψ i ɛ i , ψ i = 0.3 + 0.2x i−1 + 0.7ψ i−1 (5.40)using two different innovational distributions for ɛ i . In case 1, ɛ i is assumed to followa standardized Weibull distribution with parameter α = 1.5. In case 2, ɛ i follows a(standardized) generalized Gamma distribution with parameters κ = 1.5 andα =0.5.Figure 5.7(a) shows the time plot of the WACD(1, 1) series, whereas Figure 5.8(a)is the GACD(1, 1) series. Figure 5.9 plots the histograms of both simulated series.(a) A simulated WACD(1,1) seriesdur0 2 4 6 8 100 100 200 300 400 500(b) Standardized seriesstd-dur0 1 2 30 100 200 300 400 500Figure 5.7. A simulated WACD(1, 1) series in Eq. (5.40): (a) the original series, and (b) thestandardized series after estimation. There are 500 observations.